FA-71741 / Seismic magnitude estimation / Open access

Scalar moment from moment tensor: Frobenius half factor · case 01

Mw is 0.1 units too high for every tensor.

Verified by executionVariant 1 · 8 checks per implementationDownload source bundle ↓JSON ↗

ROOT CAUSE

The Euclidean norm is not divided by 2 under the square root.

VERIFIED REPAIR

Use sqrt(sum/2).

Unsuccessful approach: Halving outside the root is a factor of sqrt(2) too small.

Case contract

mt = [Mrr, Mtt, Mpp, Mrt, Mrp, Mtp] mantissas; tensor = mantissa * 10^exponent dyne*cm. M0 = sqrt(sum of all nine squared elements / 2) (off-diagonals appear twice), converted to N*m (1e-7). Mw = 2/3 (log10 M0 - 9.1). Flag "non-deviatoric" when |trace| > 0.1 * M0 mantissa norm, else "deviatoric". Return [Mw rounded 0.01, flag].

Why this case matters

Global CMT catalogues report Mw from the scalar moment of the moment tensor.

1 / The failure

Exit 1

"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(mt, exponent):
    mrr, mtt, mpp, mrt, mrp, mtp = mt
    s = mrr ** 2 + mtt ** 2 + mpp ** 2 + 2 * (mrt ** 2 + mrp ** 2 + mtp ** 2)
    norm = math.sqrt(s)
    m0 = norm * 10 ** exponent * 1e-7
    mw = 2.0 / 3.0 * (math.log10(m0) - 9.1)
    tr = mrr + mtt + mpp
    flag = "non-deviatoric" if abs(tr) > 0.1 * norm else "deviatoric"
    return [round(mw, 2), flag]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['CMT 0', [[1.58, -0.18, -1.4, -0.23, 1.77, 2.88], 24], [5.65, 'deviatoric']], ['CMT 1', [[1.33, 2.91, -4.19, 1.33, -1.96, -1.34], 22], [4.38, 'deviatoric']], ['CMT 2', [[-1.2, 0.01, 1.79, -1.15, 0.15, 2.83], 22], [4.29, 'non-deviatoric']], ['CMT 3', [[1.67, 0.06, -1.73, 1.49, 1.34, -2.67], 22], [4.32, 'deviatoric']], ['CMT 4', [[-1.9, -1.77, 4.27, 0.86, 2.98, 0.36], 22], [4.38, 'non-deviatoric']], ['CMT 5', [[-2.12, 2.5, -0.38, -0.41, -0.33, -2.59], 26], [6.96, 'deviatoric']], ['CMT 6', [[-0.71, 0.51, 0.2, -0.99, 2.74, 2.57], 26], [7.0, 'deviatoric']], ['CMT 7', [[1.65, 1.09, -3.64, 2.79, -1.4, 1.9], 24], [5.71, 'non-deviatoric']]], [['CMT 3', [[1.67, 0.06, -1.73, 1.49, 1.34, -2.67], 22], [4.32, 'deviatoric']], ['CMT 6', [[-0.71, 0.51, 0.2, -0.99, 2.74, 2.57], 26], [7.0, 'deviatoric']], ['CMT 7', [[1.65, 1.09, -3.64, 2.79, -1.4, 1.9], 24], [5.71, 'non-deviatoric']], ['CMT 8', [[1.74, 1.88, -3.62, -0.88, -1.61, -1.39], 22], [4.33, 'deviatoric']], ['CMT 9', [[-0.63, -0.39, 1.02, 1.72, 2.75, -2.36], 24], [5.68, 'deviatoric']], ['CMT 10', [[-2.32, -0.03, 1.45, -0.07, -0.46, -2.98], 26], [6.97, 'non-deviatoric']], ['CMT 11', [[-0.92, -0.76, 2.28, 2.62, -2.3, -1.63], 24], [5.69, 'non-deviatoric']], ['CMT 12', [[-2.44, -0.02, 2.51, 1.65, -2.68, -2.39], 24], [5.71, 'deviatoric']]], [['CMT 6', [[-0.71, 0.51, 0.2, -0.99, 2.74, 2.57], 26], [7.0, 'deviatoric']], ['CMT 11', [[-0.92, -0.76, 2.28, 2.62, -2.3, -1.63], 24], [5.69, 'non-deviatoric']], ['CMT 14', [[2.95, -1.69, -0.66, -1.55, 1.63, -1.39], 22], [4.3, 'non-deviatoric']], ['CMT 15', [[-0.88, -0.92, 1.8, 1.6, 2.66, 2.49], 22], [4.35, 'deviatoric']], ['CMT 16', [[0.43, 0.92, -1.3, -1.94, -0.05, -1.89], 24], [5.58, 'deviatoric']], ['CMT 17', [[-1.85, -0.32, 2.22, -2.09, -2.01, 2.89], 26], [7.04, 'deviatoric']], ['CMT 18', [[-0.42, 0.81, -0.39, 1.33, -2.95, 0.45], 22], [4.28, 'deviatoric']], ['CMT 19', [[-1.94, -1.33, 3.27, 1.63, 1.09, 0.68], 26], [6.96, 'deviatoric']]], [['CMT 9', [[-0.63, -0.39, 1.02, 1.72, 2.75, -2.36], 24], [5.68, 'deviatoric']], ['CMT 16', [[0.43, 0.92, -1.3, -1.94, -0.05, -1.89], 24], [5.58, 'deviatoric']], ['CMT 21', [[2.37, -2.79, 0.47, 2.71, 2.28, 2.18], 22], [4.39, 'deviatoric']], ['CMT 22', [[-0.65, -0.33, 0.98, -2.51, 0.29, 0.8], 22], [4.23, 'deviatoric']], ['CMT 23', [[1.22, -0.52, -0.7, -0.46, 0.42, 1.99], 24], [5.51, 'deviatoric']], ['CMT 24', [[-2.95, 2.42, 0.53, -0.96, 2.79, -0.69], 26], [7.01, 'deviatoric']], ['CMT 25', [[-1.12, 1.62, -1.4, -2.67, 2.0, 2.3], 24], [5.7, 'non-deviatoric']], ['CMT 26', [[-1.51, 2.8, -1.24, -0.74, 2.72, 0.61], 26], [6.98, 'deviatoric']]], [['CMT 12', [[-2.44, -0.02, 2.51, 1.65, -2.68, -2.39], 24], [5.71, 'deviatoric']], ['CMT 21', [[2.37, -2.79, 0.47, 2.71, 2.28, 2.18], 22], [4.39, 'deviatoric']], ['CMT 28', [[-2.3, -0.66, 2.96, 1.1, -2.45, 1.63], 22], [4.34, 'deviatoric']], ['CMT 29', [[-2.07, -1.28, 2.45, 2.72, -1.24, 2.82], 26], [7.05, 'non-deviatoric']], ['CMT 30', [[-1.64, -1.22, 2.86, 1.2, 0.73, -0.28], 26], [6.91, 'deviatoric']], ['CMT 31', [[0.79, 1.24, -2.03, 1.61, -0.68, 0.88], 22], [4.21, 'deviatoric']], ['CMT 32', [[-0.32, -2.12, 3.04, 2.57, 2.1, -2.5], 22], [4.39, 'non-deviatoric']], ['CMT 33', [[-2.75, -0.91, 2.76, 1.11, -2.45, 2.96], 26], [7.06, 'non-deviatoric']]]]
for label, args, expected in fixtures[N - 1]:
    check(label, solve(*args), expected)
print(json.dumps({"observations": observations, "passed": all(x["passed"] for x in observations)}, ensure_ascii=False))
raise SystemExit(0 if all(x["passed"] for x in observations) else 1)

Boundary fixture	Actual	Expected	Outcome
CMT 0	`[5.75, 'deviatoric']`	`[5.65, 'deviatoric']`	Failed
CMT 1	`[4.48, 'deviatoric']`	`[4.38, 'deviatoric']`	Failed
CMT 2	`[4.39, 'non-deviatoric']`	`[4.29, 'non-deviatoric']`	Failed
CMT 3	`[4.42, 'deviatoric']`	`[4.32, 'deviatoric']`	Failed
CMT 4	`[4.48, 'deviatoric']`	`[4.38, 'non-deviatoric']`	Failed
CMT 5	`[7.07, 'deviatoric']`	`[6.96, 'deviatoric']`	Failed
CMT 6	`[7.1, 'deviatoric']`	`[7.0, 'deviatoric']`	Failed
CMT 7	`[5.81, 'non-deviatoric']`	`[5.71, 'non-deviatoric']`	Failed

SHA-256 / b63685a03ff5fe2d3b190f4c9f1e74a280012137441f890b12e16587e16c356e

2 / The unsuccessful fix

Exit 1

"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(mt, exponent):
    mrr, mtt, mpp, mrt, mrp, mtp = mt
    s = mrr ** 2 + mtt ** 2 + mpp ** 2 + 2 * (mrt ** 2 + mrp ** 2 + mtp ** 2)
    norm = math.sqrt(s) / 2
    m0 = norm * 10 ** exponent * 1e-7
    mw = 2.0 / 3.0 * (math.log10(m0) - 9.1)
    tr = mrr + mtt + mpp
    flag = "non-deviatoric" if abs(tr) > 0.1 * norm else "deviatoric"
    return [round(mw, 2), flag]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['CMT 0', [[1.58, -0.18, -1.4, -0.23, 1.77, 2.88], 24], [5.65, 'deviatoric']], ['CMT 1', [[1.33, 2.91, -4.19, 1.33, -1.96, -1.34], 22], [4.38, 'deviatoric']], ['CMT 2', [[-1.2, 0.01, 1.79, -1.15, 0.15, 2.83], 22], [4.29, 'non-deviatoric']], ['CMT 3', [[1.67, 0.06, -1.73, 1.49, 1.34, -2.67], 22], [4.32, 'deviatoric']], ['CMT 4', [[-1.9, -1.77, 4.27, 0.86, 2.98, 0.36], 22], [4.38, 'non-deviatoric']], ['CMT 5', [[-2.12, 2.5, -0.38, -0.41, -0.33, -2.59], 26], [6.96, 'deviatoric']], ['CMT 6', [[-0.71, 0.51, 0.2, -0.99, 2.74, 2.57], 26], [7.0, 'deviatoric']], ['CMT 7', [[1.65, 1.09, -3.64, 2.79, -1.4, 1.9], 24], [5.71, 'non-deviatoric']]], [['CMT 3', [[1.67, 0.06, -1.73, 1.49, 1.34, -2.67], 22], [4.32, 'deviatoric']], ['CMT 6', [[-0.71, 0.51, 0.2, -0.99, 2.74, 2.57], 26], [7.0, 'deviatoric']], ['CMT 7', [[1.65, 1.09, -3.64, 2.79, -1.4, 1.9], 24], [5.71, 'non-deviatoric']], ['CMT 8', [[1.74, 1.88, -3.62, -0.88, -1.61, -1.39], 22], [4.33, 'deviatoric']], ['CMT 9', [[-0.63, -0.39, 1.02, 1.72, 2.75, -2.36], 24], [5.68, 'deviatoric']], ['CMT 10', [[-2.32, -0.03, 1.45, -0.07, -0.46, -2.98], 26], [6.97, 'non-deviatoric']], ['CMT 11', [[-0.92, -0.76, 2.28, 2.62, -2.3, -1.63], 24], [5.69, 'non-deviatoric']], ['CMT 12', [[-2.44, -0.02, 2.51, 1.65, -2.68, -2.39], 24], [5.71, 'deviatoric']]], [['CMT 6', [[-0.71, 0.51, 0.2, -0.99, 2.74, 2.57], 26], [7.0, 'deviatoric']], ['CMT 11', [[-0.92, -0.76, 2.28, 2.62, -2.3, -1.63], 24], [5.69, 'non-deviatoric']], ['CMT 14', [[2.95, -1.69, -0.66, -1.55, 1.63, -1.39], 22], [4.3, 'non-deviatoric']], ['CMT 15', [[-0.88, -0.92, 1.8, 1.6, 2.66, 2.49], 22], [4.35, 'deviatoric']], ['CMT 16', [[0.43, 0.92, -1.3, -1.94, -0.05, -1.89], 24], [5.58, 'deviatoric']], ['CMT 17', [[-1.85, -0.32, 2.22, -2.09, -2.01, 2.89], 26], [7.04, 'deviatoric']], ['CMT 18', [[-0.42, 0.81, -0.39, 1.33, -2.95, 0.45], 22], [4.28, 'deviatoric']], ['CMT 19', [[-1.94, -1.33, 3.27, 1.63, 1.09, 0.68], 26], [6.96, 'deviatoric']]], [['CMT 9', [[-0.63, -0.39, 1.02, 1.72, 2.75, -2.36], 24], [5.68, 'deviatoric']], ['CMT 16', [[0.43, 0.92, -1.3, -1.94, -0.05, -1.89], 24], [5.58, 'deviatoric']], ['CMT 21', [[2.37, -2.79, 0.47, 2.71, 2.28, 2.18], 22], [4.39, 'deviatoric']], ['CMT 22', [[-0.65, -0.33, 0.98, -2.51, 0.29, 0.8], 22], [4.23, 'deviatoric']], ['CMT 23', [[1.22, -0.52, -0.7, -0.46, 0.42, 1.99], 24], [5.51, 'deviatoric']], ['CMT 24', [[-2.95, 2.42, 0.53, -0.96, 2.79, -0.69], 26], [7.01, 'deviatoric']], ['CMT 25', [[-1.12, 1.62, -1.4, -2.67, 2.0, 2.3], 24], [5.7, 'non-deviatoric']], ['CMT 26', [[-1.51, 2.8, -1.24, -0.74, 2.72, 0.61], 26], [6.98, 'deviatoric']]], [['CMT 12', [[-2.44, -0.02, 2.51, 1.65, -2.68, -2.39], 24], [5.71, 'deviatoric']], ['CMT 21', [[2.37, -2.79, 0.47, 2.71, 2.28, 2.18], 22], [4.39, 'deviatoric']], ['CMT 28', [[-2.3, -0.66, 2.96, 1.1, -2.45, 1.63], 22], [4.34, 'deviatoric']], ['CMT 29', [[-2.07, -1.28, 2.45, 2.72, -1.24, 2.82], 26], [7.05, 'non-deviatoric']], ['CMT 30', [[-1.64, -1.22, 2.86, 1.2, 0.73, -0.28], 26], [6.91, 'deviatoric']], ['CMT 31', [[0.79, 1.24, -2.03, 1.61, -0.68, 0.88], 22], [4.21, 'deviatoric']], ['CMT 32', [[-0.32, -2.12, 3.04, 2.57, 2.1, -2.5], 22], [4.39, 'non-deviatoric']], ['CMT 33', [[-2.75, -0.91, 2.76, 1.11, -2.45, 2.96], 26], [7.06, 'non-deviatoric']]]]
for label, args, expected in fixtures[N - 1]:
    check(label, solve(*args), expected)
print(json.dumps({"observations": observations, "passed": all(x["passed"] for x in observations)}, ensure_ascii=False))
raise SystemExit(0 if all(x["passed"] for x in observations) else 1)

Boundary fixture	Actual	Expected	Outcome
CMT 0	`[5.55, 'deviatoric']`	`[5.65, 'deviatoric']`	Failed
CMT 1	`[4.28, 'deviatoric']`	`[4.38, 'deviatoric']`	Failed
CMT 2	`[4.19, 'non-deviatoric']`	`[4.29, 'non-deviatoric']`	Failed
CMT 3	`[4.22, 'deviatoric']`	`[4.32, 'deviatoric']`	Failed
CMT 4	`[4.28, 'non-deviatoric']`	`[4.38, 'non-deviatoric']`	Failed
CMT 5	`[6.86, 'deviatoric']`	`[6.96, 'deviatoric']`	Failed
CMT 6	`[6.9, 'deviatoric']`	`[7.0, 'deviatoric']`	Failed
CMT 7	`[5.61, 'non-deviatoric']`	`[5.71, 'non-deviatoric']`	Failed

SHA-256 / 6fa5cd17024ff303acdbe2def74076af383249cb828463b53b1be0763368c19e

3 / The verified repair

Exit 0

"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(mt, exponent):
    mrr, mtt, mpp, mrt, mrp, mtp = mt
    s = mrr ** 2 + mtt ** 2 + mpp ** 2 + 2 * (mrt ** 2 + mrp ** 2 + mtp ** 2)
    norm = math.sqrt(s / 2)
    m0 = norm * 10 ** exponent * 1e-7
    mw = 2.0 / 3.0 * (math.log10(m0) - 9.1)
    tr = mrr + mtt + mpp
    flag = "non-deviatoric" if abs(tr) > 0.1 * norm else "deviatoric"
    return [round(mw, 2), flag]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['CMT 0', [[1.58, -0.18, -1.4, -0.23, 1.77, 2.88], 24], [5.65, 'deviatoric']], ['CMT 1', [[1.33, 2.91, -4.19, 1.33, -1.96, -1.34], 22], [4.38, 'deviatoric']], ['CMT 2', [[-1.2, 0.01, 1.79, -1.15, 0.15, 2.83], 22], [4.29, 'non-deviatoric']], ['CMT 3', [[1.67, 0.06, -1.73, 1.49, 1.34, -2.67], 22], [4.32, 'deviatoric']], ['CMT 4', [[-1.9, -1.77, 4.27, 0.86, 2.98, 0.36], 22], [4.38, 'non-deviatoric']], ['CMT 5', [[-2.12, 2.5, -0.38, -0.41, -0.33, -2.59], 26], [6.96, 'deviatoric']], ['CMT 6', [[-0.71, 0.51, 0.2, -0.99, 2.74, 2.57], 26], [7.0, 'deviatoric']], ['CMT 7', [[1.65, 1.09, -3.64, 2.79, -1.4, 1.9], 24], [5.71, 'non-deviatoric']]], [['CMT 3', [[1.67, 0.06, -1.73, 1.49, 1.34, -2.67], 22], [4.32, 'deviatoric']], ['CMT 6', [[-0.71, 0.51, 0.2, -0.99, 2.74, 2.57], 26], [7.0, 'deviatoric']], ['CMT 7', [[1.65, 1.09, -3.64, 2.79, -1.4, 1.9], 24], [5.71, 'non-deviatoric']], ['CMT 8', [[1.74, 1.88, -3.62, -0.88, -1.61, -1.39], 22], [4.33, 'deviatoric']], ['CMT 9', [[-0.63, -0.39, 1.02, 1.72, 2.75, -2.36], 24], [5.68, 'deviatoric']], ['CMT 10', [[-2.32, -0.03, 1.45, -0.07, -0.46, -2.98], 26], [6.97, 'non-deviatoric']], ['CMT 11', [[-0.92, -0.76, 2.28, 2.62, -2.3, -1.63], 24], [5.69, 'non-deviatoric']], ['CMT 12', [[-2.44, -0.02, 2.51, 1.65, -2.68, -2.39], 24], [5.71, 'deviatoric']]], [['CMT 6', [[-0.71, 0.51, 0.2, -0.99, 2.74, 2.57], 26], [7.0, 'deviatoric']], ['CMT 11', [[-0.92, -0.76, 2.28, 2.62, -2.3, -1.63], 24], [5.69, 'non-deviatoric']], ['CMT 14', [[2.95, -1.69, -0.66, -1.55, 1.63, -1.39], 22], [4.3, 'non-deviatoric']], ['CMT 15', [[-0.88, -0.92, 1.8, 1.6, 2.66, 2.49], 22], [4.35, 'deviatoric']], ['CMT 16', [[0.43, 0.92, -1.3, -1.94, -0.05, -1.89], 24], [5.58, 'deviatoric']], ['CMT 17', [[-1.85, -0.32, 2.22, -2.09, -2.01, 2.89], 26], [7.04, 'deviatoric']], ['CMT 18', [[-0.42, 0.81, -0.39, 1.33, -2.95, 0.45], 22], [4.28, 'deviatoric']], ['CMT 19', [[-1.94, -1.33, 3.27, 1.63, 1.09, 0.68], 26], [6.96, 'deviatoric']]], [['CMT 9', [[-0.63, -0.39, 1.02, 1.72, 2.75, -2.36], 24], [5.68, 'deviatoric']], ['CMT 16', [[0.43, 0.92, -1.3, -1.94, -0.05, -1.89], 24], [5.58, 'deviatoric']], ['CMT 21', [[2.37, -2.79, 0.47, 2.71, 2.28, 2.18], 22], [4.39, 'deviatoric']], ['CMT 22', [[-0.65, -0.33, 0.98, -2.51, 0.29, 0.8], 22], [4.23, 'deviatoric']], ['CMT 23', [[1.22, -0.52, -0.7, -0.46, 0.42, 1.99], 24], [5.51, 'deviatoric']], ['CMT 24', [[-2.95, 2.42, 0.53, -0.96, 2.79, -0.69], 26], [7.01, 'deviatoric']], ['CMT 25', [[-1.12, 1.62, -1.4, -2.67, 2.0, 2.3], 24], [5.7, 'non-deviatoric']], ['CMT 26', [[-1.51, 2.8, -1.24, -0.74, 2.72, 0.61], 26], [6.98, 'deviatoric']]], [['CMT 12', [[-2.44, -0.02, 2.51, 1.65, -2.68, -2.39], 24], [5.71, 'deviatoric']], ['CMT 21', [[2.37, -2.79, 0.47, 2.71, 2.28, 2.18], 22], [4.39, 'deviatoric']], ['CMT 28', [[-2.3, -0.66, 2.96, 1.1, -2.45, 1.63], 22], [4.34, 'deviatoric']], ['CMT 29', [[-2.07, -1.28, 2.45, 2.72, -1.24, 2.82], 26], [7.05, 'non-deviatoric']], ['CMT 30', [[-1.64, -1.22, 2.86, 1.2, 0.73, -0.28], 26], [6.91, 'deviatoric']], ['CMT 31', [[0.79, 1.24, -2.03, 1.61, -0.68, 0.88], 22], [4.21, 'deviatoric']], ['CMT 32', [[-0.32, -2.12, 3.04, 2.57, 2.1, -2.5], 22], [4.39, 'non-deviatoric']], ['CMT 33', [[-2.75, -0.91, 2.76, 1.11, -2.45, 2.96], 26], [7.06, 'non-deviatoric']]]]
for label, args, expected in fixtures[N - 1]:
    check(label, solve(*args), expected)
print(json.dumps({"observations": observations, "passed": all(x["passed"] for x in observations)}, ensure_ascii=False))
raise SystemExit(0 if all(x["passed"] for x in observations) else 1)

Boundary fixture	Actual	Expected	Outcome
CMT 0	`[5.65, 'deviatoric']`	`[5.65, 'deviatoric']`	Passed
CMT 1	`[4.38, 'deviatoric']`	`[4.38, 'deviatoric']`	Passed
CMT 2	`[4.29, 'non-deviatoric']`	`[4.29, 'non-deviatoric']`	Passed
CMT 3	`[4.32, 'deviatoric']`	`[4.32, 'deviatoric']`	Passed
CMT 4	`[4.38, 'non-deviatoric']`	`[4.38, 'non-deviatoric']`	Passed
CMT 5	`[6.96, 'deviatoric']`	`[6.96, 'deviatoric']`	Passed
CMT 6	`[7.0, 'deviatoric']`	`[7.0, 'deviatoric']`	Passed
CMT 7	`[5.71, 'non-deviatoric']`	`[5.71, 'non-deviatoric']`	Passed

SHA-256 / 9c20a2c6e9d44089303db5ffaea5e6b0c915a48da7a89e26341843b8052ce0ca

Verification & scope

Stipulated deterministic teaching model of a seismological magnitude procedure; constants and tables are fixed by the contract and no claim of agency or standards conformance is made. This reproducer isolates one failure mechanism. Results cover the supplied fixtures. Variants within a family share a test contract and should remain grouped when constructing evaluation splits. Related mechanisms with a shared evaluation_group must also remain together; these controlled models are not independent production incidents.

Observations recorded using Python 3.12.14 at 2026-09-29T14:48:32.352474+00:00.

Case digest / d2ba4aa433780af50c4ef7aa234246a4116a44ad921954139b41ee4b85be971b